# Question: Consider the following blood inventory problem facing a hospital There

Consider the following blood inventory problem facing a hospital. There is need for a rare blood type, namely, type AB, Rh negative blood. The demand D (in pints) over any 3-day period is given by

P{D = 0} = 0.4, P{D = 1} = 0.3,

P{D = 2} = 0.2, P{D = 3} = 0.1.

The expected demand is 1 pint, since E(D) = 0.3(1) + 0.2(2) + 0.1(3) = 1. Suppose that there are 3 days between deliveries. The hospital proposes a policy of receiving 1 pint at each delivery and using the oldest blood first. If more blood is required than is on hand, an expensive emergency delivery is made. Blood is discarded if it is still on the shelf after 21 days. Denote the state of the system as the number of pints on hand just after a delivery.

Thus, because of the discarding policy, the largest possible state is 7.

(a) Construct the (one-step) transition matrix for this Markov chain.

(b) Find the steady-state probabilities of the state of the Markov chain.

(c) Use the results from part (b) to find the steady-state probability that a pint of blood will need to be discarded during a 3-day period. Because the oldest blood is used first, a pint reaches 21 days only if the state was 7 and then D = 0.

(d) Use the results from part (b) to find the steady-state probability that an emergency delivery will be needed during the 3-day period between regular deliveries.

P{D = 0} = 0.4, P{D = 1} = 0.3,

P{D = 2} = 0.2, P{D = 3} = 0.1.

The expected demand is 1 pint, since E(D) = 0.3(1) + 0.2(2) + 0.1(3) = 1. Suppose that there are 3 days between deliveries. The hospital proposes a policy of receiving 1 pint at each delivery and using the oldest blood first. If more blood is required than is on hand, an expensive emergency delivery is made. Blood is discarded if it is still on the shelf after 21 days. Denote the state of the system as the number of pints on hand just after a delivery.

Thus, because of the discarding policy, the largest possible state is 7.

(a) Construct the (one-step) transition matrix for this Markov chain.

(b) Find the steady-state probabilities of the state of the Markov chain.

(c) Use the results from part (b) to find the steady-state probability that a pint of blood will need to be discarded during a 3-day period. Because the oldest blood is used first, a pint reaches 21 days only if the state was 7 and then D = 0.

(d) Use the results from part (b) to find the steady-state probability that an emergency delivery will be needed during the 3-day period between regular deliveries.

## Answer to relevant Questions

Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 1.4. List the various financial and nonfinancial benefits that resulted from this study. Refer to Selected Reference A4 that describes an OR study done for Yellow Freight System, Inc. (a) Referring to pp. 147–149 of this article, summarize the background that led to undertaking this study. (b) Referring to p. ...In the last subsection of Sec. 29.5, the (long-run) expected average cost per week (based on just ordering costs and unsatisfied demand costs) is calculated for the inventory example of Sec. 29.1. Suppose now that the ...Consider the second version of the stock market model presented as an example in Sec. 29.2. Whether the stock goes up tomorrow depends upon whether it increased today and yesterday. If the stock increased today and ...The state of a particular continuous time Markov chain is defined as the number of jobs currently at a certain work center, where a maximum of two jobs are allowed. Jobs arrive individually. Whenever fewer than two jobs are ...Post your question